The approximation of the fourth-order partial differential equations in the class of the piecewise polynomial functions on the triangular grid

The present work determines the deviation of the piecewise cubic almost-solution of the biharmonic equation and derives the general formula (3) for its calculation. Based on this concept, we obtained an approximation of the equation. A number of numerical calculations were carried out in order to confirm the obtained formula (see pictures 1 and 2) experimentally. In general, for all selected biharmonic functions, the deviation value $\varepsilon_{\Delta \Delta}(u)$ turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with $ 25 \leq N \leq 35 $, the absolute error was about $0,0001 $, which gives an approximately asymptotic estimate of $O(h^2)$ when the partitioning step is $h \to 0$.